Check the link in the OP, the posters to that thread seemed to be of the mind that if Churchill was dead the British people and government would not be inspired continue to resist.
If anything, this scenario seems to mean that a "Fatherland" situation is most likely. The Axis is able to better prepare for the invasion of Russia for a year, plus have their vassals (France et al) also add their full economies to the war effort. Also, the Italians are likely to have a larger contribution as they won't have commitments elsewhere (like North Africa), and the Spanish might even commit more, as they won't have to worry about the reaction of the West. Who knows, maybe Finland will decided to go full tilt for Leningrad, as they won't have to worry about the West seeking vengeance and the Soviets seem more likely than ever to fall.
True, the Soviets are likely to be anticipating an invasion and plan accordingly, so they perhaps do better in 1941, stopping the Blitz much earlier. The only problem with this is that the Axis is then able to sustain a multi-season campaign much better than OTL, while the Soviets are forced to rely solely on their own resources, meaning fewer men for the front, as the civilian population is forced to turn to agriculture, mining, and manufacturing to make up for the loss of lend-lease. Otherwise mass starvation is likely, and even with the changes to the Soviet planning, it is likely that some starvation still happens anyway.
Once the Axis take Ukraine, the Soviets will be in dire straights because of the loss of food, mining, and manpower without the replacement of goods through lend lease.
The likely outcome is a Brest-Litovsk peace deal, but worse, with Stalin falling to internal house-cleaning and a wrecked Soviet Union incapable of aggression.
The Axis proceeds to ethnically cleanse large areas of their newly-won territory, but things slow down once Hitler dies and the lack of a replacement with the necessary gravitas to hold the Nazi party together (Göring is about 5 years from a heart attack or diabetes doing him in, Himmler is hated by everyone, Bormann also has enemies and is a faceless bureaucrat, Goebbels is a joke). Germany probably falls to a stooge who is supported by the army, which then either opts for bringing the Monarchy back, or running the country indefinitely. Or a civil war between the SS and Wehrmacht.